On the Kazhdan–Lusztig order on cells and families

Abstract

We consider the set Irr(W) of (complex) irreducible characters of a finite Coxeter group W. The Kazhdan–Lusztig theory of cells gives rise to a partition of Irr(W) into “families” and to a natural partial order ${\le }_{\mathcal{L}\mathcal{R}}$ on these families. Following an idea of Spaltenstein, we show that ${\le }_{\mathcal{L}\mathcal{R}}$ can be characterised (and effectively computed) in terms of standard operations in the character ring of W. If, moreover, W is the Weyl group of an algebraic group G, then ${\le }_{\mathcal{L}\mathcal{R}}$ can be interpreted, via the Springer correspondence, in terms of the closure relation among the “special” unipotent classes of G.